Average fragment carrying a coloured tip

Question

Some 9-inch glass rods each have a blue mark on one end and a red mark on the other. After a fall, several rods shattered into three pieces at random break points. Among those rods, what is the average length of the piece bearing the blue mark?

Short answer

3 inches. With two independent uniform break points, the three piece lengths are exchangeable, so they share the same mean. Since the lengths sum to 9 inches, each piece — including the blue-tipped end — averages 3 inches.

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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